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Introduction CHAPTER 1 Introduction 1. Overview Quantum field theory has been wildly successful as a framework for the study of high-energy particle physics. In addition, the ideas and techniques of quantum field theory have had a profound influence on the development of mathematics. There is no broad consensus in the mathematics community, however, as to what quantum field theory actually is. This book develops another point of view on perturbative quantum field theory, based on a novel axiomatic formulation. Most axiomatic formulations of quantum field theory in the literature start from the Hamiltonian formulation of field theory. Thus, the Segal [Seg99] axioms for field theory propose that one assigns a Hilbert space of states to a closed Riemannian man- ifold of dimension d 1, and a unitary operator between Hilbert spaces to a d dimen- − sional manifold with boundary. In the case when the d dimensional manifold is of the form M [0, t], we should view the corresponding operator as time evolution. × The Haag-Kastler [Haa92] axioms also start from the Hamiltonian formulation, but in a slightly different way. They take as the primary object not the Hilbert space, but rather a C! algebra, which will act on a vacuum Hilbert space. I believe that the Lagrangian formulation of quantum field theory, using Feyn- man’s sum over histories, is more fundamental. The axiomatic framework devel- oped in this book is based on the Lagrangian formalism, and on the ideas of low- energy effective field theory developed by Kadanoff [Kad66], Wilson [Wil71], Polchin- ski [Pol84] and others. 1.1. The idea of the definition of quantum field theory I use is very simple. Let us assume that we are limited, by the power of our detectors, to studying physical 7 8 1. INTRODUCTION phenomena that occur below a certain energy, say Λ. The part of physics that is visible to a detector of resolution Λ we will call the low-energy effective field theory. This low-energy effective field theory is succinctly encoded by the energy Λ version of the Lagrangian, which is called the low-energy effective action Se f f [Λ]. The notorious infinities of quantum field theory only occur if we consider phenom- ena of arbitrarily high energy. Thus, if we restrict attention to phenomena occurring at energies less than Λ, we can compute any quantity we would like in terms of the effective action Se f f [Λ]. If Λ# < Λ, then the energy Λ# effective field theory can be deduced from knowl- edge of the energy Λ effective field theory. This leads to an equation expressing the e f f e f f scale Λ# effective action S [Λ#] in terms of the scale Λ effective action S [Λ]. This equation is called the renormalization group equation. If we do have a continuum quantum field theory (whatever that is!) we should, in particular, have a low-energy effective field theory for every energy. This leads to our definition : a continuum quantum field theory is a sequence of low-energy effective actions Se f f [Λ], for all Λ < , which are related by the renormalization group flow. In addition, we require that the Se f f [Λ] satisfy a locality axiom, which says that the effective actions Se f f [Λ] become∞ more and more local as Λ . → This definition aims to be as parsimonious as possible. The∞only assumptions I am making about the nature of quantum field theory are the following: (1) The action principle: physics at every energy scale is described by a La- grangian, according to Feynman’s sum-over-histories philosophy. (2) Locality: in the limit as energy scales go to infinity, interactions between fields occur at points. 1.2. In this book, I develop complete foundations for perturbative quantum field theory in Riemannian signature, on any manifold, using this definition. The first significant theorem I prove is an existence result: there are as many quan- tum field theories, using this definition, as there are Lagrangians. Let me state this theorem more precisely. Throughout the book, I will treat h¯ as a formal parameter; all quantities will be formal power series in h¯ . Setting h¯ to zero amounts to passing to the classical limit. 1. OVERVIEW 9 Let us fix a classical action functional Scl on some space of fields E , which is as- sumed to be the space of global sections of a vector bundle on a manifold M1. Let T (n)(E , Scl) be the space of quantizations of the classical theory that are defined mod- ulo h¯ n+1. Then, 1.2.1 Theorem. T (n+1)(E , Scl) T (n)(E , Scl) → is a torsor for the abelian group of Lagrangians under addition (modulo those Lagrangians which are a total derivative). Thus, any quantization defined to order n in h¯ can be lifted to a quantization de- fined to order n + 1 in h¯ , but there is no canonical lift; any two lifts differ by the addition of a Lagrangian. If we choose a section of each torsor T (n+)(E , Scl) T (n)(E , Scl) we find an → isomorphism T ( )(E , Scl) = series Scl + h¯ S(1) + h¯ 2 S(2) + ∼ · · · where each S(i) is a local∞ functional, that is, a functional which can be written as the in- tegral of a Lagrangian. Thus, this theorem allows one to quantize the theory associated to any classical action functional Scl. However, there is an ambiguity to quantization: at each term in h¯ , we are free to add an arbitrary local functional to our action. 1.3. The main results of this book are all stated in the context of this theorem. In Chapter 4, I give a definition of an action of the group R>0 on the space of theories on Rn. This action is called the local renormalization group flow, and is a funda- mental part of the concept of renormalizability developed by Wilson and others. The n action of group R>0 on the space of theories on R simply arises from the action of this group on Rn by rescaling. The coefficients of the action of this local renormalization group flow on any par- ticular theory are the functions of that theory. I include explicit calculations of the function of some simple theories, including the 4 theory on R4. This local renormalization group flow leads to a concept of renormalizability. Fol- lowing Wilson and others, I say that a theory is perturbatively renormalizable if it has “critical” scaling behaviour under the renormalization group flow. This means that 1The classical action needs to satisfy some non-degeneracy conditions 10 1. INTRODUCTION the theory is fixed under the renormalization group flow except for logarithmic cor- rections. I then classify all possible renormalizable scalar field theories, and find the expected answer. For example, the only renormalizable scalar field theory in four di- mensions, invariant under isometries and under the transformation , is the → − 4 theory. In Chapter 5, I show how to include gauge theories in my definition of quantum field theory, using a natural synthesis of the Wilsonian effective action picture and the Batalin-Vilkovisky formalism. Gauge symmetry, in our set up, is expressed by the requirement that the effective action Se f f [Λ] at each energy Λ satisfies a certain scale Λ Batalin-Vilkovisky quantum master equation. The renormalization group flow is compatible with the Batalin-Vilkovisky quantum master equation: the flow from scale Λ to scale Λ# takes a solution of the scale Λ master equation to a solution to the scale Λ# equation. I develop a cohomological approach to constructing theories which are renormal- izable and which satisfy the quantum master equation. Given any classical gauge theory, satisfying the classical analog of renormalizability, I prove a general theorem allowing one to construct a renormalizable quantization, providing a certain cohomol- ogy group vanishes. The dimension of the space of possible renormalizable quantiza- tions is given by a different cohomology group. In Chapter 6, I apply this general theorem to prove renormalizability of pure Yang- Mills theory. To apply the general theorem to this example, one needs to calculate the cohomology groups controlling obstructions and deformations. This turns out to be a lengthy (if straightforward) exercise in Gel’fand-Fuchs Lie algebra cohomology. Thus, in the approach to quantum field theory presented here, to prove renormal- izability of a particular theory, one simply has to calculate the appropriate cohomology groups. No manipulation of Feynman graphs is required. 2. Functional integrals in quantum field theory Let us now turn to giving a detailed overview of the results of this book. First I will review, at a basic level, some ideas from the functional integral point of view on quantum field theory. 2. FUNCTIONAL INTEGRALS IN QUANTUM FIELD THEORY 11 2.1. Let M be a manifold with a metric of Lorentzian signature. We will think of M as space-time. Let us consider a quantum field theory of a single scalar field : M R. → The space of fields of the theory is C (M). We will assume that we have an action functional of the form ∞ S( ) = L ( )(x) x M ! ∈ where L ( ) is a Lagrangian. A typical Lagrangian of interest would be L ( ) = 1 (D +m2) + 1 4 − 2 4! where D is the Lorentzian analog of the Laplacian operator. A field C (M, R) can describes one possible history of the universe in this ∈ simple model. ∞ Feynman’s sum-over-histories approach to quantum field theory says that the uni- verse is in a quantum superposition of all states C (M, R), each weighted by iS( )/h¯ ∈ e .
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